Born-Infeld theory with higher derivatives

We present new models of nonlinear electromagnetism which satisfy the Noether-Gaillard-Zumino current conservation and are, therefore, self-dual. The new models differ from the Born-Infeld–type models in that they deform the Maxwell theory starting with terms like $\lambda (\partial F{)}^4$. We provide a recursive algorithm to find all higher-order terms in the action of the form ${\lambda }^n{\partial }^{4n}{F}^{2n+2}$, which are necessary for the $U(1)$ duality current conservation. We use one of these models to find a self-dual completion of the $\lambda (\partial F{)}^4$ correction to the open string action. We discuss the implication of these findings for the issue of UV finiteness of $\mathcal{N}=8$ supergravity.